Extreme Points of Gram Spectrahedra of Binary Forms

The Gram spectrahedron Gram(f) of a form f with real coefficients is a compact affine-linear section of the cone of psd symmetric matrices. It parametrizes the sum of squares decompositions of f , modulo orthogonal equivalence. For f a sufficiently general positive binary form of arbitrary degree, we show that Gram(f) has extreme points of all ranks in the Pataki range. We also calculate the dimension of the set of rank r extreme points, for any r . Moreover, we determine the pairs of rank two extreme points for which the connecting line segment is an edge of Gram(f) . The proof of the main result relies on a purely algebraic fact of independent interest: Whenever d,r≥ 1 are integers with ( [ r+1; 2 ]) ≤ 2d+1 , there exists a length r sequence f_1,… ,f_r of binary forms of degree d for which the ( [ r+1; 2 ]) pairwise products f_if_j , i≤ j , are linearly independent.